Description
In this lab, we used data collected from a calorimetry experiment to determine the specific
heat of an unknown sample. We also used error propagation techniques to find the error in
the calculated specific heat. We then used our calculated specific heat to identify the unknown
sample using a given table of candidate materials and their properties. I calculated the specific
heat of the unknown sample to be 0.248 ± 0.000268 [J/kg*K]. The calculated value of specific
heat most closely matches the specific heat of Tellurium Copper Alloy, with a discrepancy of
0.013 [J/kg*K]. Thus, it was determined that the known sample was most likely Tellurium
Copper (Alloy 145).
I. Nomenclature
mc = mass of calorimeter
ms = mass of sample
cc = specific heat capacity of calorimeter
cs = specific heat capacity of sample
T2 = equilibrium temperature of calorimeter and sample
T1 = initial temperature of sample
T0 = initial temperature of calorimeter
∆U = change in internal energy of system
U2 = initial internal energy of system
U1 = final internal energy of system
II. Introduction
T
his project uses concepts from calorimetry to find the specific heat of an unknown sample. Calorimetry is defined as a
basic technique for measuring thermodynamic properties. In a typical calorimetry experiment, an unknown sample is
placed inside a well-insulated container filled with a liquid with known properties. As heat is transferred from the sample
to the known liquid, it becomes possible to determine the properties of the sample. With regards to this specific experiment, water was chosen to be the known liquid, and a standard purchased calorimeter was used as the insulated container.
The formula used to calculate the specific heat of the sample is as follows:
cs =
mccc(T2 − T0)
ms(T1 − T2)
(1)
With regards to each term in the above equation:
mc and ms can be measured by some type of mass balance
cc should be given by the manufacturer of the calorimeter
T1 can be obtained from the temperature of the water in which the sample is heated
T0 can be obtained from the temperature profile of the calorimeter
T2 can be obtained from the temperature profile of the calorimeter
Note that the because the container used for calorimetry isn’t perfectly insulated, it is necessary to use a procedure
provided by the manufacturer of the calorimeter that uses the method of least squares fit and extrapolation to better
approximate T2, T1, and T0. More specifically, T0 is approximated by fitting a line to the pre-sample temperature
∗Undergraduate Student, Aerospace Engineering, 5515 Northern Lights Drive
collected during the first 10 minutes of the experiment. Then the line is extrapolated forward to the time the sample was
added: 11 minutes into the experiment. A second line is then fitted from the maximum temperature reading to the last
temperature reading. This line is then extrapolated back to the time when the sample was added: 11 minutes into the
experiment. These two extrapolated temperature values are then averaged and the second line of best fit is extrapolated
backwards to the time corresponding to this average temperature value. The resulting extrapolated temperature value is
T2, the final temperature of the calorimeter and sample at equilibrium. These new extrapolated approximates for T0 and
T2 are then plugged into formula derived above for calculating the specific heat of the unknown sample, together with
mc, ms, cc, and T1.
In addition, it is important to note that each of the of the terms present in equation 1 has uncertainty, and that these
uncertainties must be propagated using the techniques covered in class in order to find the uncertainty in the calculated
specific heat of the sample.
The general error propagation formula was used to calculate the error in the specific heat of the sample. This
formula, along with all of the partial derivatives used in it, are listed below.
σcs =
s
(
∂cs
∂mc
∗ σmc)
2 + (
∂cs
∂T2
∗ σT2)
2 + (
∂cs
∂T1
∗ σT1)
2 + (
∂cs
∂ms
∗ σms)
2 + (
∂cs
∂T0
∗ σT0)
2
(2)
∂cs
∂mc
=
cc(−T0 + T2)
ms(T1 − T2)
(3)
∂cs
∂T2
=
ccmc
ms(T1 − T2)
+
ccmc(−T0 + T2)
ms(T1 − T2)
2
(4)
∂cs
∂T0
= −
ccmc
ms(T1 − T2)
(5)
∂cs
∂T1
= −
ccmc(−T0 + T2)
ms(T1 − T2)
2
(6)
∂cs
∂ms
= −
ccmc(−T0 + T2)
m2
s
(T1 − T2)
(7)
III. Experimental Method
1) Examine the calorimeter. Note that the calorimeter is made of aluminum with an exact specific heat of 0.214
cal/(g*C).
2) The sample is weighed multiple times to determine an average mass.
3) A thermocouple with software cold-junction compensation and ITLL LabStations was used to take temperature
readings of the aluminum calorimeter.
4) The thermocouple is placed into the hold provided and secured with high temperature cotton before the insulation
cap is replaced.
5) The sample is immersed in boiling water for about 10 minutes until is in equilibrium with the boiling water.
6) At the same time, the sample is immersed in the boiling water, the temperature measuring software is initiated. It
takes samples every second.
7) The sample is then removed from the boiling water using tongs. It is shaken and then quickly placed inside the
calorimeter. The calorimeter is then sealed.
8) The temperature measuring software runs for about 10 minutes before the experiment ends.
9) The program is terminated and the data is saved
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Fig. 1 Temperature Profile of Calorimeter with Least Square Best Fit Lines.
Fig. 2 Temperature Profile of Boiling Water with Least Square Best Fit Lines.
IV. Results
The specific heat of the sample was found to be approximately 0.248 J/gK with an error of approximately 0.000172
J/gK. Comparing this value with the those in the provided table of candidate materials properties shows that while the
specific heat of the sample doesn’t exactly match up with any of the 4 provided materials, it most closely matches the
specific heat of Tellurium Copper (Alloy 145), which has an specific heat of 0.261 J/gK. For a more explicit comparison,
consider that the difference between the specific heat of the sample and the Tellurium Copper is a mere 0.013 J/gK. On
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the other hand, the difference between the specific heat of the sample and Zn (0.402 J/gK), Pb (0.129 J/gK), and Al (0.9
J/gK) is 0.154 J/gK, 0.119 J/gK, and 0.652 J/gK, respectively.
V. Discussion
In this case, the deviation between the measured specific heat of the sample and the specific heat of Tellurium
Copper is significant, as the specific heat of Tellurium Copper does not fall within the error bounds of the measured
specific heat of the sample. This indicates that the errors of the parameters used in my calculation of the specific heat
of the sample need to be larger than they are now. Looking at each of the parameters in equation 1, the errors in the
temperature measurements T0, T1, T2 were all calculated using the methods taught in class. Assuming these methods
were used properly, the only to obtain a larger error in these values is to increase the range of temperature measurements
used to produce the lines of best fit, thereby introducing greater variation between the values. Perhaps a more simple
method to increase the error in the calculated specific heat of the sample is to increase the error assigned to the mass of
the calorimeter and the sample. This increase in error may be justified if there were human errors made when measuring
the mass of the calorimeter or sample. For example, it is possible that the scale used wasn’t calibrated properly, or
not enough mass measurements were taken before calculating the average mass. Finally, perhaps a more practical
justification for the discrepancy between the measured and accepted specific heat is that the method used to approximate
the values of T0,T1,T2 is flawed. For example, the measured specific heat would be greater if the difference between T2
and T0 or T1 and T2 increased. That means that if the approximated value of T0 were lower and T1 were higher, the
discrepancy between the measured and accepted specific heat values could be reduced. As such, if this experiment were
conducted using a more accurate calorimeter with better insulation, we would likely obtain better results.
VI. Conclusion
The identity of the sample is most likely Tellurium Copper (Alloy 145). While the calculated specific heat of the
sample does deviate significantly from the accepted specific heat, it was still possible to identify the the sample with a
great degree of certainty. This is because the discrepancy between the specific heat of the sample and Tellurium Copper
is far lower than the discrepancy between the specific heat of the sample and Zn-Cu-Ti, Pb, or the discrepancy between
the specific heat of the sample and 6063-T1 Al. Thus, it is reasonable to deem this experiment a success.
References
Jackson, J., “ASEN 2012 Project 1 Calorimetry,” Oct. 2017.
Jackson, J., “Calorimetry Experimental Procedure,” Oct. 2017.
Jackson, J., “Calorimetry Additional Data,” Oct. 2017.
Jackson, J., “Candidate Materials Properties,” Oct. 2017.
Jackson, J., “ASEN 2012 Project 1 Calorimetry,” Oct. 2017.
Appendix
The complete derivation of the formula used to calculate the specific of the sample is as follows:
∆U = 0 (8)
U2 − U1 = 0 (9)
(mcuc2 + msus2) − (mcuc1 + msus1) = 0 (10)
ms cs(T1 − T2) = mccc(T2 − T0) (11)
cs =
mccc(T2 − T0)
ms(T1 − T2)
(12)
Explicit Use of Engineering Method for Algorithm Development
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• Problem: I need use the method of least squares and the error propagation formulas learned in class to identify an
unknown sample by calculating its specific heat.
• Knowns: Data that contain measurements of the temperature at different times of the calorimeter and boiling
water bath. The mass of the sample and of the calorimeter and their uncertainties.
• Find: The specific heat of the sample and its uncertainty using the data provided.
• Assumptions: The mass and uncertainties provided are correct. The sample is added to the calorimeter 10 minutes
after the data started to be collected. It is valid to use all error propagation and other techniques covered in class.
• Sketch: See algorithm diagrams below.
• Fundamentals: See equations and other concepts presented in the introduction.
• Alternatives: Could use a better calorimeter, eliminating the need to extrapolate and approximate. Could use a
mass spectrometer or other devices to use the metals composition to determine its identity. Could also carry out
chemistry experiments or visually compare it with other materials to test its properties and help identify it.
• Steps:
1) Parse data
2) Find T0
3) Find error in T0
4) Find T2
5) Find error in T2
6) Find T1
7) Find error in T1
8) Find the specific heat of the sample
9) Find the error in the specific heat of the sample
10) Plot the temperature profile and least squares best fit lines for the calorimeter and boiling water
11) Print findings to output file
• Check: I have verified that my value for the specific heat of the sample and the error in the specific heat of the
sample matches that of my peers.
• Makes Sense?: I believe so. There wasn’t very much variation in the points used to generate all 3 lines of best fit.
In addition, the error in the masses was small. Thus, it makes sense for the error in the specific heat to be small.
The calculated value of the specific heat is similar to that of Tellurium Copper, which seems reasonable.
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Fig. 3 Main Program Algorithm Flow Chart
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Fig. 4 Subroutine 1: readinput Algorithm Flow Chart
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Fig. 5 Subroutine 2: findT0 Algorithm Flow Chart
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Fig. 6 Subroutine 3: findsigT0 Algorithm Flow Chart
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Fig. 7 Subroutine 4: findT2 Algorithm Flow Chart
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Fig. 8 Subroutine 5: findsigT2 Algorithm Flow Chart
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Fig. 9 Subroutine 6: findT2 Algorithm Flow Chart
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Fig. 10 Subroutine 7: findsigT2 Algorithm Flow Chart
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Fig. 11 Subroutine 8: findcs Algorithm Flow Chart
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Fig. 12 Subroutine 9: findsigcs Algorithm Flow Chart
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Fig. 13 Subroutine 10: createplots Algorithm Flow Chart
Fig. 14 Subroutine 11: writeoutput Algorithm Flow Chart
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